Branching Laws Research Articles

Branching laws for discrete series of some affine symmetric spaces

In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $\Pi$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representation $\Pi$ and the corresponding Arthur-Vogan packets to smaller orthogonal groups; these are inspired by the Gross-Prasad conjectures.

75 Years of International Law-Making at the United Nations

For the last 75 years, the UN has been placed at the centre of international law-making. The Organization can be considered both as a place where international law is discussed, made and interpreted by its Member States and as a proper actor, with its own international legal personality, voice and practice, engaged in the creation and implementation of international law. This article considers the extraordinary position of the UN in providing a unique contribution to the development, codification and implementation of international law in branches ranging from the law of treaties to the legal principles governing the protection and preservation of the marine environment or the criminal accountability for graves violations of international humanitarian law. On its 75th anniversary, the Organization has demonstrated its flexibility and adaptability to the changing priorities and concerns of the international community and facilitated the commitment of its Member States to multilateralism and the principles enshrined in the Charter signed on 26 June 1945 in San Francisco. Indeed, the outlook for the next 25 years of international law-making at the UN looks brighter than it may at first appear.

Branching laws for classical groups: the non-tempered case

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

Branching problems in reproducing kernel spaces

For a semisimple Lie group G satisfying the equal-rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In our work here we study some of the branching laws for discrete series when restricted to a subgroup H of the same type by combining classical results with recent work of Kobayashi; in particular, we prove discrete decomposability under Harish-Chandra’s condition of cusp form on the reproducing kernel. We show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators.

A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].

Some applications of theta correspondence to branching laws

In this paper, we will use the local theta correspondences for the dual pair (Sp(W),O(V)) to investigate some branching law problems.

Optimal survival strategy for branching Brownian motion in a Poissonian trap field

Nous étudions un mouvement brownien branchant $Z$ ayant une loi de branchement générique et évoluant dans $\mathbb{R}^{d}$, où se trouve un champ de pièges poissonniens. Chaque piège est constitué d’une boule de rayon constant. Les pièges sont durs, au sens où le processus est tué instantanément dès qu’il pénètre dans l’un des pièges. Nous nous concentrons sur deux cas particuliers de champs poissonniens, un champ uniforme et un champ décroissant radialement, et nous considérons un environnement annealed. En utilisant des résultats classiques sur la convergence de la vitesse du mouvement brownien branchant, nous établissons des résultats annealed précis sur la taille de la population décrite par $Z$, conditionnellement à ce qu’il évite l’ensemble des pièges et reste en vie jusqu’au temps $t$. Les résultats sont formulés de sorte que chacun d’entre eux donne une ‘stratégie optimale de survie’ pour $Z$. En corollaires de ces résultats, nous démontrons l’optimalité de plusieurs autres stratégies concernant le support de $Z$ jusqu’au temps $t$ et la taille et la position de clairières dans $\mathbb{R}^{d}$. Nous démontrons également un résultat sur le temps d’atteinte d’un seul piège par un système branchant (Lemme 1), qui pourra aussi être utile dans un cadre totalement générique. Au passage, nous apportons une réponse à plusieurs questions ouvertes formulées dans (Markov Process. Related Fields 9 (2003) 363–389).

Noncommutative Boundaries Arising in Dynamics and Representations of the Cuntz Relations

In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum probability, there has also been much research which seeks to study fractal and multiresolution structures with the use of certain systems of noncommutative operators; noncommutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these noncommutative algebras are the Cuntz algebras; denoted N for the number of isometry generators. N is at least 2. Our focus is on the representations of We aim to develop new noncommutative tools, involving both representation theory and stochastic processes. They serve to connect these parallel developments.In outline, boundaries, Poisson, or Martin, are certain measure spaces (often associated to random walk models), designed to encode the asymptotic behavior, e.g., how trajectories diverge when the number of steps goes to infinity. We stress that our present boundaries (commutative or noncommutative) are purely measure-theoretical objects. Although, as we show, in some cases our boundaries may be compared with more familiar topological boundaries.

Branching problems for semisimple Lie groups and reproducing kernels

For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for these when restricted to a subgroup H of the same type by combining the classical results with the recent work of T. Kobayashi. We analyze aspects of having differential operators being symmetry-breaking operators; in particular, we prove in the so-called admissible case that every symmetry breaking (H-map) operator is a differential operator. We prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernels. Our techniques are based on realizing discrete series representations as kernels of elliptic invariant differential operators.

Triple product p-adic L-functions for balanced weights

We construct p-adic triple product L-functions that interpolate (square roots of) central critical L-values in the balanced region. Thus, our construction complements that of Harris and Tilouine. There are four central critical regions for the triple product L-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three p-adic L-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region: we produce the corresponding p-adic L-function. Our triple product p-adic L-function arises as p-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of p-adic period integrals is showing that these branching laws vary in a p-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.

Competition and Banks’ Cost of Equity Capital: Evidence from Relatively Exogenous Differences in Regulation

We examine the effects of geographic deregulation on banks’ cost of equity (COE) using changes in interstate bank branching laws over the post–Riegle-Neal period (1994:Q4–2016:Q4). We find strong evidence that deregulation increases banks’ COE. This is driven primarily by active acquirers, rather than those subject to increased competition from these acquirers. We identify higher risk-taking as an important channel for these findings. The results support our new acquisitions–fragility view, rather than the traditional competition–fragility view. Results are robust to instrumental variables, accounting for secular trends, and alternative explanations.

On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements

Given a branching random walk on $${\mathbb {R}}$$ started from the origin, where the tail of the branching law decays at least exponentially fast and the offspring number is at least one, let $$Z_n(\cdot )$$ be the counting measure which counts the number of individuals at the n-th generation located in a given set. Under some mild conditions, it is known (Biggins in Stoch. Process. Appl. 34:255–274, 1990) that for any interval $$A\subset {\mathbb {R}}$$ , $$\frac{Z_n(\sqrt{n}A)}{Z_n({\mathbb {R}})}$$ converges a.s. to $$\nu (A)$$ , where $$\nu $$ is the standard Gaussian measure. In this work, we investigate the convergence rates of $$\begin{aligned} {\mathbb {P}}\left( \frac{Z_n(\sqrt{n}A)}{Z_n({\mathbb {R}})}-\nu (A)>\Delta \right) , \end{aligned}$$ for $$\Delta \in (0, 1-\nu (A))$$ . We consider both the Schroder case, where the offspring number could be one, and the Bottcher case, where the offspring number is at least two.

SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over $\mathbb{Q}$ of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use $R=T$ theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from $\text{GSp}(4)$ to $U(4)$. In that case we obtain a corollary for abelian surfaces.

Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures’ laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form Vn−φlogn+oP(logn), where Vn is a function of the environment that behaves as a random walk and φ>0 is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.

Market Actor's Regime in Spanish Jurisdiction from the Perspective of Sustainability Law

In the first attempts to apply the concept of sustainability to legal systems, sustainability was understood as a unidimensional category aimed at protecting the environment. Nowadays, however, as will be seen in this paper, sustainability cannot be understood or reached if it is not the result of the intertwined harmony of all three internal dimensions which this concept is considered to include: the economic dimension, the environmental dimension, and the social dimension. We should not only desire, but rather demand that one of the goals of countries´ jurisdiction should be to promote sustainability. In this context, we can discuss the concept Sustainability Law, understood as the regulations which address circumstances that facilitate and promote sustainability. Given that sustainability, with its three dimensions, is a transversal concept, Sustainability Law must never become an isolated sector of legal systems, disconnected from the rest. The whole legal system must be inspired by sustainability as one of its multiple functions. Therefore, with a broad and all-embracing point of view of countries´ legal systems, it could be stated that a State has Sustainability Law, or, even better, Sustainable Law, if the regulations of that territory as a whole comply with the defence and promotion of sustainability. The purpose of this study is to analyse the state of Sustainability Law in branches of the Spanish legal system which are essential when attempting to achieve sustainable and responsible trade, especially focusing on the regulation of market actors.

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation $\pi$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $\nu$ occurring in the restriction $\pi|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $\pi$ and $\nu$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits $O^G$ of a semisimple Lie group $G$ corresponding to highest weight modules of scalar type. We prove that the Corwin--Greenleaf number $\sharp(O^G \cap pr^{-1}(O^H))/H$ is either zero or one for any $H$-coadjoint orbit $O^H$, whenever $(G,H)$ is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits $O^H$ with nonzero Corwin-Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as "classical limits" of the multiplicity-free branching laws of holomorphic discrete series representations (T.Kobayashi [Progr.Math.2007]).

A Research of Optimization of Water-gate System Based on Three Bilateral Clearance Cooperation Policies at Container Terminal

The following passage mainly focus on the knowledge of clearance cooperation at inland city and seaside city, as well as the achievement of administrative bilateral department exchange information. Based on the cooperative strategies of water-gate system optimization at container terminal, the department will not only recognize its supervision bilaterally, but also they are allowed to achieve a mutual law enforcement. Besides, the passage will also talk about the establishment of clearance cooperation of both cross-divisions and cross-region from both inland city and seaside city. Optimizing clearance procedure and identifying bilateral monitor result by standardizing internal supervision in the same division as well as enforcing the law in different branches, the government can push to a management institution which is not only appropriate to Chinese currency, but also it is competitive enough to meet globalization.

Does Competition Reduce Racial Discrimination in Lending?

This paper examines whether increases in bank competition reduce discriminatory practices in mortgage lending. Lenders are significantly less likely to approve black applicants' loan applications despite facing similar credit risk. However, following the relaxation of interstate bank branching laws in the 1990s, increases in local lending competition reduced the approval differential between potential white and black borrowers by roughly one quarter. The reduction was driven both by incumbent lenders altering lending policies to avoid losing market share and by the entry of new banks. The results suggest strong complementaries between direct regulation and the competition mechanism. In particular, direct regulation is effective against large lenders where statistical proof problems are less severe, while competition provides incentives to smaller, harder to regulate lenders.

Coronary fractional flow reserve measurements of a stenosed side branch: a computational study investigating the influence of the bifurcation angle.

BackgroundCoronary hemodynamics and physiology specific for bifurcation lesions was not well understood. To investigate the influence of the bifurcation angle on the intracoronary hemodynamics of side branch (SB) lesions computational fluid dynamics simulations were performed.MethodsA parametric model representing a left anterior descending—first diagonal coronary bifurcation lesion was created according to the literature. Diameters obeyed fractal branching laws. Proximal and distal main branch (DMB) stenoses were both set at 60 %. We varied the distal bifurcation angles (40°, 55°, and 70°), the flow splits to the DMB and SB (55 %:45 %, 65 %:35 %, and 75 %:25 %), and the SB stenoses (40, 60, and 80 %), resulting in 27 simulations. Fractional flow reserve, defined as the ratio between the mean distal stenosis and mean aortic pressure during maximal hyperemia, was calculated for the DMB and SB (FFRSB) for all simulations.ResultsThe largest differences in FFRSB comparing the largest and smallest bifurcation angles were 0.02 (in cases with 40 % SB stenosis, irrespective of the assumed flow split) and 0.05 (in cases with 60 % SB stenosis, flow split 55 %:45 %). When the SB stenosis was 80 %, the difference in FFRSB between the largest and smallest bifurcation angle was 0.33 (flow split 55 %:45 %). By describing the ΔPSB−QSB relationship using a quadratic curve for cases with 80 % SB stenosis, we found that the curve was steeper (i.e. higher flow resistance) when bifurcation angle increases (ΔP = 0.451*Q + 0.010*Q2 and ΔP = 0.687*Q + 0.017*Q2 for 40° and 70° bifurcation angle, respectively). Our analyses revealed complex hemodynamics in all cases with evident counter-rotating helical flow structures. Larger bifurcation angles resulted in more pronounced helical flow structures (i.e. higher helicity intensity), when 60 or 80 % SB stenoses were present. A good correlation (R2 = 0.80) between the SB pressure drop and helicity intensity was also found.ConclusionsOur analyses showed that, in bifurcation lesions with 60 % MB stenosis and 80 % SB stenosis, SB pressure drop is higher for larger bifurcation angles suggesting higher flow resistance (i.e. curves describing the ΔPSB−QSB relationship being steeper). When the SB stenosis is mild (40 %) or moderate (60 %), SB resistance is minimally influenced by the bifurcation angle, with differences not being clinically meaningful. Our findings also highlighted the complex interplay between anatomy, pressure drops, and blood flow helicity in bifurcations.

Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs

Rankin–Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of $$SL(2,\mathbb {R})$$ . We address a general problem to find explicit formulae for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose, we use a new method (F-method) developed in Kobayashi and Pevzner (Sel. Math. New Ser., (2015). doi: 10.1007/s00029-15-0207-9 ) and based on the algebraic Fourier transform for generalized Verma modules. The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulae of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin–Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.